Spectral homotopy analysis method for solving nonlinear Volterra integro differential equations

In this paper we proposed Spectral Homotopy analysis method to solve nonlinear Volterra integro-differential equations. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method and exact analytical solutions

Numerical Solution for Linear Fredholm Integro-Differential Equation Using Touchard Polynomials

 تم تقديم طريقة جديدة تستند الى متعددة حدود تشارد للحل العددي لمعادلات فريدهولم التفاضلية التكاملية من المرتبة الاولى والنوع الثاني مع الشرط.  تم الحصول ببساطة على مشتقة متعددة حدود تشارد وتكاملها. واعطيت دقة الطريقة المقدمة وثبتت قابلية تطبيقها ببعض الامثلة العددية.  تتم مقارنة النتائج التي تم الحصول عليها مع النتائج المعروفة  الاخرى.A new method based on the Touchard polynomials (TPs) was presented for the numerical solution of the linear Fredholm integro-differential equation (FIDE) of the first order and second kind with condition. The derivative and integration of the (TPs) were simply obtained. The convergence analysis of the presented method was given and the applicability was proved by some numerical examples. The results obtained in this method are compared with other known results.

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad

A new numerical technique based on Chelyshkov polynomials for solving two-dimensional stochastic It\^o-Volterra Fredholm integral equation

In this paper, a two-dimensional operational matrix method based on Chelyshkov polynomials is implemented to numerically solve the two-dimensional stochastic It\^o-Volterra Fredholm integral equations. These equations arise in several problems such as an exponential population growth model with several independent white noise sources. In this paper a new stochastic operational matrix has been derived first time ever by using Chelyshkov polynomials. After that, the operational matrices are used to transform the It\^o-Volterra Fredholm integral equation into a system of linear algebraic equations by using Newton cotes nodes as collocation point that can be easily solved. Furthermore, the convergence and error bound of the suggested method are well established. In order to illustrate the effectiveness, plausibility, reliability, and applicability of the existing technique, two typical examples have been presented

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Numerical solutions of integral equations by using CAS wavelets

Wavelets are mathematical functions that cut up data into different frequency components and study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelets are developed independently in the field of mathematics, quantum physics, electrical engineering, and seismic geology. Interchange between this field during the last ten years have led to many new wavelet application such as image compression, turbulence, human vision, radar and earthquake prediction. In this we introduce a numerical method of solving integral equation by using CAS wavelets. This method is method upon CAS wavelet approximations. The properties of CAS wavelets are first presented. CAS wavelet approximations methods are then utilized to reduce the integral equations to the solution of algebraic equations

Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation

This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the $L_2$ and ${L_\infty }$ error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations